If \(n\in N\)  then \(\Omega (n)\)  usually denotes the number of
primes dividing \(n\)  counting their multiplicities (see [1]}).
In [2]} and [3]}, independent families of events have been
studied for finite probability spaces with uniform distribution.
Eisenberg and Ghosh [3]} show that the number of nontrivial
independent events in such spaces cannot be more than \(\Omega (m)\) 
where \(m\)  is the cardinality of the space. This result should be
seen in view of the known fact (see Problem 50, Section 4.1 in
[5]}) that if \(A_1\) , \(A_2\) , ...., \(A_n\)  are independent
non-trivial events of a sample space \(X\)  then \(|X|\ge 2^n\) . One can
observe that in general \(\Omega (m)\)  is considerably smaller than
\(\log _2 m\) . It is worth mentioning that according to [6]} the
first paper to deal to this problem in uniform finite probability
spaces is [7]}. In their paper, Shiflett and Shultz [7]}
raise the question of the existence of spaces with no non-trivial
independent pairs, called dependent probability spaces. A
space containing non-trivial independent events is called independent. For uniform distributed probability spaces \(X\) , as a
result of the work in [3]} and [2]}, \(X\)  is dependent if
\(|X|\)  a prime number and independent if \(|X|\)  is composite. For
denumerable sets \(X\)  one can see the construction given in
[6]} or look at the Example 1.1 in [12]}. For our spaces,
the Example 1.1 does not apply and in fact, we will construct
explicitly lots of independent sets.
